# Wave Motion

## Wave propagation

 Wave motion is the transfer of energy without transfer of matter. Waves can occur as mechanical waves or electromagnetic radiation.

Electromagnetic radiation propagates by the interaction of perpendicular electric and magnetic fields, and so can travel in a vacuum. Mechanical waves propagate by the interaction of elements in a physical medium. This means that mechanical waves can only travel in a physical medium where the elements of the medium are able to interact and transmit the wave. A source of disturbance is also required. For example, a string is a physical medium which can transmit waves because the string particles are connected together by intermolecular forces. An initial disturbance could be a person flicking one end of the rope, which would produce a "bump" travelling along the string.

A single disturbance propagating through a medium (such as the "bump" above) is called a pulse. A pulse has definite height and propagates through the medium with definite speed. The pulse shape does not change. A wave is formed when a periodic disturbance produces many pulses in succession.

### Transverse and Longitudinal Waves

• Transverse waves have the disturbance perpendicular to the direction of travel. As the wave moves left to right, particles in the medium are displaced up and down. Examples include waves on a string and electromagnetic radiation.
• Longitudinal waves are where the disturbance moves parallel to the direction of travel. This causes the elements to compress together and then stretch out as the wave passes. Sound waves are an important example of this.
• Complex waves are combinations of transverse and longitudinal waves.

Only energy is transferred by the wave, regardless of its type. The elements of the medium return to their original position once the wave passes.

### Travelling Waves

A wave, or pulse, may be represented graphically. The graph below shows a pulse on a string. The vertical axis represents the transverse displacement and the horizontal axis describes the location along the string of each element.

The transverse displacement is a function of time, pulse velocity, and particle location on the string. This is described in the wave function below.

y(x,t) = f(x - vt)

If time (t) is fixed, then the function describes the waveform (shape) at that point in time. If location (x) was fixed, then the function would describe the vertical displacement of that point over time.

The simplest form of travelling wave is the sinusoidal wave (same curve as y=sinθ). Repeated, consecutive pulses form a sinusoidal wave which travels through the medium.

### Wave Equations

• The crest of a wave is the position of maximum displacement (highest point on a peak).
• The amplitude (A) is the height of these crests.
• The wavelength (λ) is the distance between two crests (or between any two identical points on adjacent waves).
• The period (T) is the time between two crests (or between any two identical points on adjacent waves). It is the time for one full wave to pass.
• The frequency (f) is the number of crests which pass per second.

Frequency is the inverse of period: f = 1/T

• Angular wave number (or wave number), k, is defined as k=(2π)/λ
• Angular frequency (ω) is defined as ω=2πf=(2π)/T.

A wave travelling to the right, at time t, with speed v, has the following equation:

y(x,t) = Asin[((2π)/λ)(x-vt)]

This may be written more simply using wave number and angular frequency.

y(x,t) = Asin(kx-ωt)

For a wave travelling to the left, replace (kx-ωt) with (kx+ωt). Note that the above equations apply to waves for which x=0 and y=0 when t=0 (the initial displacement at the origin was 0). If this was not the case, a phase constant (φ) may be used, shifting the wave equation to fit the initial conditions. The phase constant may be calculated by comparing the wave equation with the waveform at t=0.

y(x,t) = Asin(kx-ωt+φ)

### Wave Speed

 Any wave's speed may be found using the following equation:

v = fλ = ω/k

For a pulse on a string, the wave speed may be calculated from string tension (T) and mass per unit length (μ):

v = sqrt(T/μ)

The transverse speed and acceleration of wave particles may be found by differentiating the wave equation with respect to time. Note that transverse speed is not wave speed.

vy = -ωAcos(kx-ωt) ay = -ω2Asin(kx-ωt)

The transverse motion of each particle in the wave is simple harmonic motion.

## Reflection and Transmission

 When a wave strikes a boundary between two media, part of the wave is reflected and part of it is transmitted. Conservation of energy means that the sum of reflected and transmitted wave energies must equal the energy of the original wave. (Note that we are ignoring energy loss to heat etc).

### Reflection of Pulses on a String

 When a pulse on a string reaches the end of the string, the pulse is completely reflected (like a wave striking a completely reflective boundary). If the sting end is fixed, the pulse is inverted (upside down). If the string end is loose, the pulse is not inverted.

When a pulse on a string reaches a different string type, some of the pulse will be reflected and some will be transmitted (like a wave striking a boundary between two media). If the pulse travels from a light string to a heavy string, the reflected pulse is inverted. If the pulse travels from heavy to light, the reflected pulse is not inverted. In both cases, the transmitted pulse is not inverted.

Each new pulse has lower amplitude than the original, because the sum of reflected and transmitted pulse energies equals the original pulse energy.

## Power and Intensity

 Waves transmit energy. The wave source performs work on the wave, which puts energy into the wave. This energy propagates along the wave and is released when the wave interacts with an object. For example, a wave on a string carries kinetic energy and can transfer this energy to objects attached to the string.

The power of a wave (rate of energy transfer) is proportional to the square of amplitude and the square of angular frequency. For a wave on a string,

P = ½μω2A2v, where μ is mass per unit length, ω is angular frequency, A is amplitude, and v is wave speed.

## Interference

 Unlike particles, two waves can exist in the same place at the same time. When this happens, they interfere. The waves are not destroyed or altered, but while they are in the same place they combine according to the principle of superposition.

### Superposition

Superposition is when two (or more) interfering waves combine to form a resultant wave. The resultant wave is just the sum of the two waveforms. The resultant wave equation may be calculated by adding together the interfering wave equations.

• If the two waves interfere to produce a larger resultant wave, constructive interference is occurring.
• If the two waves interfere to produce a smaller (or 0) resultant wave, destructive interference is occurring.

For two interfering sinusoidal waves with the same frequency, wavelength and amplitude, resultant wave may be found as follows.

y1(x,t) = Asin(kx-ωt) y2(x,t) = Asin(kx-ωt+φ)

y1 + y2 = yres

yres(x,t) = 2Acos(φ/2)sin(kx-ωt+φ/2)

In this case, constructive interference occurs when φ=0, since the amplitude becomes 2A. Destructive interference occurs when φ=nπ (n is an odd integer)as the amplitude is then 0.

### Interference in Sound Waves

Sound waves with identical frequency, amplitude and wavelength will interfere if they originate from different sources. The different path lengths from the observer to the sound source mean that the waves from each source will arrive out of phase and so interfere.

The difference in path length (Δr) may be expressed in terms of phase difference (φ):

Δr = (φλ)/(2π)

This gives the following conditions for constructive and destructive interference:

• Constructive interference occurs when Δr = (2n)(λ/2)
• Destrutive interference occurs when Δr = (2n+1)(λ/2)
• ('n' is any integer.)

### Standing Waves

If two waves with the same amplitude, frequency and wavelength travel in opposite directions through a medium, they will interfere to form a standing wave.

y1(x,t) = Asin(kx-ωt) y2(x,t) = Asin(kx+ωt)

yres = (2Asin(kx))cos(ωt)

This wave does not travel, since there is no (kx-ωt) term. Instead, it oscillates in place. The cos(ωt) term describes the oscillations with respect to time (frequency ω). The amplitude of these oscillations is explained below.

• The amplitude of the individual interfering waves is A
• The amplitude of the standing (resultant) wave is 2A
• The amplitude of any particular element on the wave is 2Asin(kx). This means that the location (x) of a particle determines the amplitude of its oscillation. This is evident from the animation below, where some parts of the wave oscillate with high amplitude and some parts don't oscillate at all.
• The points which do not oscillate are called nodes (zero amplitude).
• Nodes occur when x=(nλ)/2 (n is any integer)
• The distance between adjacent nodes is λ/2.
• Antinodes are points of maximum amplitude (amplitude = 2A).
• Antinodes occur when x=(nλ)/4 (n is any odd integer)
• The distance between adjacent antinodes is λ/2.
• The distance between a node and an adjacent antinode is λ/4.

#### Standing Waves on a String

If a string is fixed at both ends, these points must be nodes, since they can only have zero displacement. This is a boundary condition on the vibration of the string. Depending on the boundary condition, a string (or other medium) will have a set of normal modes which it can vibrate at. Each mode has a characteristic (specific) frequency. Since only specific frequencies are permitted, the wave is said to be quantized.

Once the boundary condition is known, the normal modes can be found by keeping the distance between nodes and antinodes as λ/4. Below are the first three normal modes of vibration for a string fixed at both ends.

• The first normal mode has the longest wavelength
• There is one antinode in the middle
• λ=2L
• n=1
• λ=L
• n=2
• Each vibrational mode has one more antinode than the one before.
• λ=(2L)/3
• n=3

The normal wavelengths of a string fixed at both ends are given by the following equation:

λn = 2L/n where 'n' is the nth mode of oscillation.

• If the string was loose at one end, that end would form an antinode, rather than a node.
• Other oscillating systems behave in a similar way, such as sound waves a pipe.

#### Harmonics

• The lowest vibrational mode (largest λ) is the fundamental frequency.
• On a string instrument, the fundamental frequency is the frequency of the open string. It can be altered by changing the string length (pressing down on a fret) or tension (tuning the string).
• The frequencies of the normal modes are integer multiples of the fundamental frequency.
• The higher normal modes are known as harmonics.
• On a string instrument, harmonics are played by holding the string in place at node locations (achieved by resting a finger lightly on the string).

## End

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